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The Fredkin gate is CSWAP gate. Given a control register in $0$ or $1$, the gate does nothing or swaps two target registers respectively.

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Is there a higher dimensional version of this gate? I have one d-dimensional control register (this takes values from the set $S = \{1, 2, ... , d\}$) and $d$ target registers. If the control register takes on value $i\in S$, then I would like to swap the first target register with the $i^{th}$ target register.

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  • $\begingroup$ @MarkS aren't all swaps (controlled or otherwise) reversible trivially? Or did I misunderstand you? $\endgroup$ Nov 8 '21 at 18:31
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The question seems to be about a higher-dimensional control, and not about higher-dimensional targets. For $d=4$, would the following circuit work?

HighDimensionalControl

Furthermore although this wasn't the question asked, one could instead consider another CSWAP gate that has one qubit control, and higher-dimensional qudits that are swapped conditioned on the qubit. One example also for $d=4$ might be below. The four bottom qubits are to be thought of as two pairs of $4$-dimensional qudits.

Qudit CSWAPs

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